Methods in studying qualitative properties of fractional equations

TL;DR

This paper systematically reviews various effective methods for analyzing the qualitative properties of solutions to fractional equations, serving as a comprehensive handbook for researchers in the field.

Contribution

It introduces and compares multiple methods, including extension, moving planes, blow-up, and regularity techniques, for studying fractional equations' solutions.

Findings

Comparison of method strengths and limitations

Illustrative examples of method applications

A comprehensive overview of qualitative analysis techniques

Abstract

In this paper, we systematically review a series of effective methods for studying the qualitative properties of solutions to fractional equations. Beginning with the pioneering extension method and the method of moving planes in integral forms, we introduce a variety of direct methods, including the direct method of moving planes, the method of moving spheres, blow-up and rescaling techniques, the sliding method, regularity lifting, and approaches for interior and boundary regularity estimates. To elucidate the core ideas behind these methods, we employ simple examples that demonstrate how they can be applied to investigate qualitative properties of solutions. We also provide a comparative discussion of their respective strengths and limitations. It is our hope that this paper will serve as a useful handbook for researchers engaged in the study of fractional equations.